Question

an increasing function f(x) has f(0) = 10 and f(5) = 18. which of the following is the best estimate of ∫50()? (a) 40 (b) 70 (c) 100 (d) 8 (e) 18

Answer 1

The best estimate of the integral from 0 to 5 for an increasing function with given values is calculated using the average value over the interval, leading to an answer of 70.

Step-by-step explanation:

The student is asked to estimate the integral of an increasing function from 0 to 5, with known function values at f(0) = 10 and f(5) = 18. Since the function is increasing, the area under the curve between x = 0 and x = 5 can be estimated using the average value of the function over the interval times the length of the interval. The average value of the function can be taken as the midpoint of f(0) and f(5), which is (10 + 18) / 2 = 14. The length of the interval is 5. Therefore, the estimated integral, or the area under the curve, is 14 multiplied by 5, which is 70.

Answer 2

So, the best estimate for
`\( \int_(0)^(5) f(x) \, dx \)` is (b) 70.

Since
`\( f(x) \)` is an increasing function and
`\( f(0) = 10 \) and \( f(5) = 18 \)`, we can estimate the definite integral
`\( \int_(0)^(5) f(x) \, dx \)` as the area under the curve between
`\( x = 0 \) and \( x = 5 \)`. The given options are:

(a) 40

(b) 70

(c) 100

(d) 8

(e) 18

Since
`\( f(x) \)` is increasing, the estimate of the definite integral can be obtained by finding the area of a trapezoid formed by the points
`\((0, 10)\) and \((5, 18)\)`.

`\[ \text{Area of trapezoid} = (1)/(2) \cdot (f(0) + f(5)) \cdot \text{base} \]`

`\[ \text{Area of trapezoid} = (1)/(2) \cdot (10 + 18) \cdot 5 = (28)/(2) \cdot 5 = 70 \]`